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zero point; continuation method; $C^{1}$-homotopy; surjerctive implicit function theorem; proper mapping; compact mapping; coercive mapping; Fredholm mapping
Some sufficient conditions are provided that guarantee that the difference of a compact mapping and a proper mapping defined between any two Banach spaces over $\mathbb {K}$ has at least one zero. When conditions are strengthened, this difference has at most a finite number of zeros throughout the entire space. The proof of the result is constructive and is based upon a continuation method.
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